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In this research, we examine a diffusive predator–prey model with spatial memory. We begin by checking that the suggested model has a unique solution that is boundedness. The stability of each equilibrium is then examined. Local and global stability as well as bifurcations are investigated in the non-delayed model at stationary equilibrium. Then, we investigate the Hopf bifurcation using the delay as the bifurcation parameter. In order to back up our theoretical findings, we then give some numerical simulations. [ABSTRACT FROM AUTHOR]

In epidemics, waning immunity is common after infection or vaccination of individuals. Immunity levels are highly heterogeneous and dynamic. This work presents an immuno-epidemiological model that captures the fundamental dynamic features of immunity acquisition and wane after infection or vaccination and analyzes mathematically its dynamical properties. The model consists of a system of first order partial differential equations, involving nonlinear integral terms and different transfer velocities. Structurally, the equation may be interpreted as a Fokker-Planck equation for a piecewise deterministic process. However, unlike the usual models, our equation involves nonlocal effects, representing the infectivity of the whole environment. This, together with the presence of different transfer velocities, makes the proved existence of a solution novel and nontrivial. In addition, the asymptotic behavior of the model is analyzed based on the obtained qualitative properties of the solution. An optimal control problem with objective function including the total number of deaths and costs of vaccination is explored. Numerical results describe the dynamic relationship between contact rates and optimal solutions. The approach can contribute to the understanding of the dynamics of immune responses at population level and may guide public health policies. [ABSTRACT FROM AUTHOR]

Most water-borne disease models ignore the advection of water flows in order to simplify the mathematical analysis and numerical computation. However, advection can play an important role in determining the disease transmission dynamics. In this paper, we investigate the long-term dynamics of a periodic reaction–advection–diffusion schistosomiasis model and explore the joint impact of advection, seasonality and spatial heterogeneity on the transmission of the disease. We derive the basic reproduction number R 0 and show that the disease-free periodic solution is globally attractive when R 0 < 1 whereas there is a positive endemic periodic solution and the system is uniformly persistent in a special case when R 0 > 1 . Moreover, we find that R 0 is a decreasing function of the advection coefficients which offers insights into why schistosomiasis is more serious in regions with slow water flows. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reaction–advection–diffusion dengue fever model with seasonal developmental durations and intrinsic incubation periods. Firstly, we establish the well-posedness of the model. Secondly, we define the basic reproduction number ℜ 0 for this model and show that ℜ 0 is a threshold parameter: if ℜ 0 < 1 , then the disease-free periodic solution is globally attractive; if ℜ 0 > 1 , the system is uniformly persistent. Thirdly, we study the global attractivity of the positive steady state when the spatial environment is homogeneous and the advection of mosquitoes is ignored. As an example, we use the model to investigate the dengue fever transmission case in Guangdong Province, China, and explore the impact of model parameters on ℜ 0 . Our findings indicate that ignoring seasonality may underestimate ℜ 0 . Additionally, the spatial heterogeneity of transmission may increase the risk of disease transmission, while the increase of seasonal developmental durations, intrinsic incubation periods and advection rates can all reduce the risk of disease transmission. [ABSTRACT FROM AUTHOR]

The design of optimized non-pharmaceutical interventions (NPIs) is critical to the effective control of emergent outbreaks of infectious diseases such as SARS, A/H1N1 and COVID-19 and to ensure that numbers of hospitalized cases do not exceed the carrying capacity of medical resources. To address this issue, we formulated a classic SIR model to include a close contact tracing strategy and structured prevention and control interruptions (SPCIs). The impact of the timing of SPCIs on the maximum number of non-isolated infected individuals and on the duration of an infectious disease outside quarantined areas (i.e. implementing a dynamic zero-case policy) were analyzed numerically and theoretically. These analyses revealed that to minimize the maximum number of non-isolated infected individuals, the optimal time to initiate SPCIs is when they can control the peak value of a second rebound of the epidemic to be equal to the first peak value. More individuals may be infected at the peak of the second wave with a stronger intervention during SPCIs. The longer the duration of the intervention and the stronger the contact tracing intensity during SPCIs, the more effective they are in shortening the duration of an infectious disease outside quarantined areas. The dynamic evolution of the number of isolated and non-isolated individuals, including two peaks and long tail patterns, have been confirmed by various real data sets of multiple-wave COVID-19 epidemics in China. Our results provide important theoretical support for the adjustment of NPI strategies in relation to a given carrying capacity of medical resources. [ABSTRACT FROM AUTHOR]

This paper is concerned with threshold dynamics of a degenerate diffusive HBV infection model with DNA-containing capsids in heterogeneous environment. We firstly address the existence of global solutions, uniform and ultimate boundedness of solutions, asymptotic smoothness of semiflows and existence of a connected global attractor for the diffusive model. Then, we identify the basic reproduction number R 0 and establish a threshold-type result for the disease eradication or uniform persistence when R 0 ≤ 1 or R 0 > 1 , respectively. Especially, when R 0 > 1 and the diffusion rate of capsids or the diffusion rate of virions is zero, we further show that the model admits a unique infection steady state which is globally attractive. Our results indicate that the pathogen can be eliminated by limiting the mobility of the capsids or virions. [ABSTRACT FROM AUTHOR]

Considering the importance of time delay phenomenon in disease transmission and the interdependence of time delay and spatial location, a reaction-diffusion SIR epidemic model with nonlocal time delay (or time-space time delay) is proposed and the travelling wave solutions are discussed. Specifically, we define the basic reproduction number $ \mathcal {R}_0 $ R0 and the critical wave speed $ c^* $ c∗. For every wave speed $ c\geq c^* $ c≥c∗, the existence of travelling wave solutions is studied by using the upper–lower solutions, the fixed-point theorem and some limit techniques when $ \mathcal {R}_0 \gt 1 $ R0>1. The nonexistence of travelling waves when $ \mathcal {R}_0 \gt 1 $ R0>1 for any $ 0 \lt c \lt c^* $ 0c>0 is also proved. Finally, the influence of time-delay on disease propagation, especially the critical wave speed, is discussed through analysis and simulations. [ABSTRACT FROM AUTHOR]

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR—model and on a susceptible-exposed-infected-recovered SEIR—model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero. [ABSTRACT FROM AUTHOR]

This study addresses COVID-19 testing as a nonlinear sampling problem, aiming to uncover the dependence of the true infection count in the population on COVID-19 testing metrics such as testing volume and positivity rates. Employing an artificial neural network, we explore the relationship among daily confirmed case counts, testing data, population statistics, and the actual daily case count. The trained artificial neural network undergoes testing in in-sample, out-of-sample, and several hypothetical scenarios. A substantial focus of this paper lies in the estimation of the daily true case count, which serves as the output set of our training process. To achieve this, we implement a regularized backcasting technique that utilize death counts and the infection fatality ratio (IFR), as the death statistics and serological surveys (providing the IFR) as more reliable COVID-19 data sources. Addressing the impact of factors such as age distribution, vaccination, and emerging variants on the IFR time series is a pivotal aspect of our analysis. We expect our study to enhance our understanding of the genuine implications of the COVID-19 pandemic, subsequently benefiting mitigation strategies. [ABSTRACT FROM AUTHOR]

The use of therapeutic agents is a critical option to manage wildlife disease, but their implementation is usually spatially constrained. We seek to expand knowledge around the effectiveness of management of environmentally-transmitted Sarcoptes scabiei on a host population, by studying the effect of a spatially constrained treatment regime on disease dynamics in the bare-nosed wombat Vombatus ursinus. A host population of wombats is modelled using a system of non-linear partial differential equations, a spatially-varying treatment regime is applied to this population and the dynamics are studied over a period of several years. Treatment could result in mite decrease within the treatment region, extending to a lesser degree outside, with significant increases in wombat population. However, the benefits of targeted treatment regions within an environment are shown to be dependent on conditions at the start (endemic vs. disease free), as well as on the locations of these special regions (centre of the wombat population or against a geographical boundary). This research demonstrates the importance of understanding the state of the environment and populations before treatment commences, the effects of re-treatment schedules within the treatment region, and the transient large-scale changes in mite numbers that can be brought about by sudden changes to the environment. It also demonstrates that, with good knowledge of the host-pathogen dynamics and the spatial terrain, it is possible to achieve substantial reduction in mite numbers within the target region, with increases in wombat numbers throughout the environment. [ABSTRACT FROM AUTHOR]

Diagnostic delay for TB infected individuals and the lack of TB vaccines for adults are the main challenges to achieve the goals of WHO by 2050. In order to evaluate the impacts of diagnostic delay and vaccination for adults on prevalence of TB, we propose an age-structured model with latent age and infection age, and we incorporate Mycobacterium TB in the environment and vaccination into the model. Diagnostic delay is indicated by the age of infection before receiving treatment. The threshold dynamics are established in terms of the basic reproduction number R 0 . When R 0 < 1 , the disease-free equilibrium is globally asymptotically stable, which means that TB epidemic will die out; When R 0 = 1 , the disease-free equilibrium is globally attractive; there exists a unique endemic equilibrium and the endemic equilibrium is globally attractive when R 0 > 1 . We estimate that the basic reproduction number R 0 = 0.5320 (95% CI (0.3060, 0.7556)) in Jiangsu Province, which means that TB epidemic will die out. However, we find that the annual number of new TB cases by 2050 is 1,151 (95%CI: (138, 8,014)), which means that it is challenging to achieve the goal of WHO by 2050. To this end, we evaluate the possibility of achieving the goals of WHO if we start vaccinating adults and reduce diagnostic delay in 2025. Our results demonstrate that when the diagnostic delay is reduced from longer than four months to four months, or 20% adults are vaccinated, the goal of WHO in 2050 can be achieved, and 73,137 (95%CI: (23,906, 234,086)) and 54,828 (95%CI: (15,811, 206,468)) individuals will be prevented from being infected from 2025 to 2050, respectively. The modeling approaches and simulation results used in this work can help policymakers design control measures to reduce the prevalence of TB. [ABSTRACT FROM AUTHOR]

Emerging and re-emerging pathogens are latent threats in our society with the risk of killing millions of people worldwide, without forgetting the severe economic and educational backlogs. From COVID-19, we learned that self isolation and quarantine restrictions (confinement) were the main way of protection till availability of vaccines. However, abrupt lifting of social confinement would result in new waves of new infection cases and high death tolls. Here, inspired by how an extracellular solution can make water move into or out of a cell through osmosis, we define confinement tonicity. This can serve as a standalone measurement for the net direction and magnitude of flows between the confined and deconfined susceptible compartments. Numerical results offer insights on the effects of easing quarantine restrictions. [ABSTRACT FROM AUTHOR]

A significant proportion of malaria infections in humans exhibit no symptoms, but it is a reservoir for maintaining malaria transmission. A time periodic reaction-diffusion model for malaria spread is introduced in this paper, incorporating spatial heterogeneity, incubation periods, symptomatic and asymptomatic carriers. This paper introduces the concept of the basic reproduction number R 0 , which is defined as the spectral radius of the next generation operator, and we present some preliminary results by elementary analysis. The threshold dynamic behavior analysis shows that when R 0 < 1 , the disease is extinct, and when R 0 > 1 , the disease is persistent. We investigate the case of constant system parameters, focusing on the global asymptotic stability of the disease-free steady state when R 0 = 1 . In the numerical simulation section, we validate the theoretical results obtained, and then use elasticity analysis methods to explore the influence of parameters on the output solution. In addition, sensitivity analysis of the basic reproduction number under homogeneous conditions indicates direction of controlling malaria transmission. And several control measures are evaluated in the following steps. [ABSTRACT FROM AUTHOR]

COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, R 0 , which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when R 0 < 1 , that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate ν 1 leads to oscillations. When ν 1 increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation. [ABSTRACT FROM AUTHOR]

This study formulates a host–pathogen model driven by cross-diffusion to examine the effect of chemotaxis on solution dynamics and spatial structures. The negative binomial incidence mechanism is incorporated to illustrate the transmission process by pathogens. In terms of the magnitude of chemotaxis, the global solvability of the model is extensively studied by employing semigroup methods, loop arguments, and energy estimates. In a limiting case, the necessary conditions for chemotaxis-driven instability are established regarding the degree of chemotactic attraction. Spatial aggregation may occur along strong chemotaxis in a two-dimensional domain due to solution explosion. We further observe that spatial segregation appears for short-lived free pathogens in a one-dimensional domain, whereas strong chemotactic repulsion homogenizes the infected hosts and thus fails to segregate host groups effectively. [ABSTRACT FROM AUTHOR]

Decline of the dissolved oxygen in the ocean is a growing concern, as it may eventually lead to global anoxia, an elevated mortality of marine fauna and even a mass extinction. Deoxygenation of the ocean often results in the formation of oxygen minimum zones (OMZ): large domains where the abundance of oxygen is much lower than that in the surrounding ocean environment. Factors and processes resulting in the OMZ formation remain controversial. We consider a conceptual model of coupled plankton–oxygen dynamics that, apart from the plankton growth and the oxygen production by phytoplankton, also accounts for the difference in the timescales for phyto- and zooplankton (making it a “slow-fast system”) and for the implicit effect of upper trophic levels resulting in density dependent (nonlinear) zooplankton mortality. The model is investigated using a combination of analytical techniques and numerical simulations. The slow-fast system is decomposed into its slow and fast subsystems. The critical manifold of the slow-fast system and its stability is then studied by analyzing the bifurcation structure of the fast subsystem. We obtain the canard cycles of the slow-fast system for a range of parameter values. However, the system does not allow for persistent relaxation oscillations; instead, the blowup of the canard cycle results in plankton extinction and oxygen depletion. For the spatially explicit model, the earlier works in this direction did not take into account the density dependent mortality rate of the zooplankton, and thus could exhibit Turing pattern. However, the inclusion of the density dependent mortality into the system can lead to stationary Turing patterns. The dynamics of the system is then studied near the Turing bifurcation threshold. We further consider the effect of the self-movement of the zooplankton along with the turbulent mixing. We show that an initial non-uniform perturbation can lead to the formation of an OMZ, which then grows in size and spreads over space. For a sufficiently large timescale separation, the spread of the OMZ can result in global anoxia. [ABSTRACT FROM AUTHOR]

Malaria is a vector-borne disease that exacts a grave toll in the Global South. The epidemiology of Plasmodium vivax, the most geographically expansive agent of human malaria, is characterised by the accrual of a reservoir of dormant parasites known as hypnozoites. Relapses, arising from hypnozoite activation events, comprise the majority of the blood-stage infection burden, with implications for the acquisition of immunity and the distribution of superinfection. Here, we construct a novel model for the transmission of P. vivax that concurrently accounts for the accrual of the hypnozoite reservoir, (blood-stage) superinfection and the acquisition of immunity. We begin by using an infinite-server queueing network model to characterise the within-host dynamics as a function of mosquito-to-human transmission intensity, extending our previous model to capture a discretised immunity level. To model transmission-blocking and antidisease immunity, we allow for geometric decay in the respective probabilities of successful human-to-mosquito transmission and symptomatic blood-stage infection as a function of this immunity level. Under a hybrid approximation—whereby probabilistic within-host distributions are cast as expected population-level proportions—we couple host and vector dynamics to recover a deterministic compartmental model in line with Ross-Macdonald theory. We then perform a steady-state analysis for this compartmental model, informed by the (analytic) distributions derived at the within-host level. To characterise transient dynamics, we derive a reduced system of integrodifferential equations, likewise informed by our within-host queueing network, allowing us to recover population-level distributions for various quantities of epidemiological interest. In capturing the interplay between hypnozoite accrual, superinfection and acquired immunity—and providing, to the best of our knowledge, the most complete population-level distributions for a range of epidemiological values—our model provides insights into important, but poorly understood, epidemiological features of P. vivax. [ABSTRACT FROM AUTHOR]

Multiple infections enable the recombination of different strains, which may contribute to viral diversity. How multiple infections affect the competition dynamics between the two types of strains, the wild and the immune escape mutant, remains poorly understood. This study develops a novel mathematical model that includes the two strains, two modes of viral infection, and multiple infections. For the representative double-infection case, the reproductive numbers are derived and global stabilities of equilibria are obtained via the Lyapunov direct method and theory of limiting systems. Numerical simulations indicate similar viral dynamics regardless of multiplicities of infections though the competition between the two strains would be the fiercest in the case of quadruple infections. Through sensitivity analysis, we evaluate the effect of parameters on the set-point viral loads in the presence and absence of multiple infections. The model with multiple infections predict that there exists a threshold for cytotoxic T lymphocytes (CTLs) to minimize the overall viral load. Weak or strong CTLs immune response can result in high overall viral load. If the strength of CTLs maintains at an intermediate level, the fitness cost of the mutant is likely to have a significant impact on the evolutionary dynamics of mutant viruses. We further investigate how multiple infections alter the viral dynamics during the combination antiretroviral therapy (cART). The results show that viral loads may be underestimated during cART if multiple-infection is not taken into account. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number R 0 < 1 . For R 0 > 1 , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics. [ABSTRACT FROM AUTHOR]

The increase in antibiotic resistance in recent years, mainly due to the non-rational use of antibiotics, is one of the most important global public health threats. In this paper, we propose a mathematical dynamic random model describing the antibiotic resistance evolution of a bacteria and where antibiotic consumption is included is the main driving force in the resistance increase. The random model is solved using the Random Variable Transformation technique and is applied to study the case of Acinetobacter baumannii bacterium resistant to the antibiotic colistin in Valencia, Spain. Using the Multi-Objective Particle Swarm Optimization algorithm, the model has been calibrated with the A. baumannii colistin-resistance and colistin consumption data series. With the optimal model, four possible 7-year future scenarios with different antibiotic consumption trends have been simulated. The model results show how reducing antibiotic consumption does not easily stop the increase in resistance. [ABSTRACT FROM AUTHOR]

Lyme disease is the most common vector-borne disease in the United States impacting the Northeast and Midwest at the highest rates. Recently, it has become established in southeastern and south-central regions of Canada. In these regions, Lyme disease is caused by Borrelia burgdorferi, which is transmitted to humans by an infected Ixodes scapularis tick. Understanding the parasite-host interaction is critical as the white-footed mouse is one of the most competent reservoir for B. burgdorferi. The cycle of infection is driven by tick larvae feeding on infected mice that molt into infected nymphs and then transmit the disease to another susceptible host such as mice or humans. Lyme disease in humans is generally caused by the bite of an infected nymph. The main aim of this investigation is to study how diapause delays and demographic and seasonal variability in tick births, deaths, and feedings impact the infection dynamics of the tick-mouse cycle. We model tick-mouse dynamics with fixed diapause delays and more realistic Erlang distributed delays through delay and ordinary differential equations (ODEs). To account for demographic and seasonal variability, the ODEs are generalized to a continuous-time Markov chain (CTMC). The basic reproduction number and parameter sensitivity analysis are computed for the ODEs. The CTMC is used to investigate the probability of Lyme disease emergence when ticks and mice are introduced, a few of which are infected. The probability of disease emergence is highly dependent on the time and the infected species introduced. Infected mice introduced during the summer season result in the highest probability of disease emergence. [ABSTRACT FROM AUTHOR]

In this paper, we study a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number R 0 and show that the infection-free steady state is globally asymptotically stable when R 0 ≤ 1 , while the disease is uniformly persistent when R 0 > 1 . In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of R 0 and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results, conduct sensitivity analysis, and investigate the impact of drugs on curtailing the spread of the viruses. [ABSTRACT FROM AUTHOR]

Ontogenic resistance has been described for many plant-pathogen systems. Conversely, coffee leaf rust, a major fungal disease that drastically reduces coffee production, exhibits a form of ontogenic susceptibility, with a higher infection risk for mature leaves. To take into account stage-dependent crop response to phytopathogenic fungi, we developed an SEIR-U epidemiological model, where U stands for spores, which differentiates between young and mature leaves. Based on this model, we also explored the impact of ontogenic resistance on the sporulation rate. We computed the basic reproduction number R 0 , which classically determines the stability of the disease-free equilibrium. We identified forward and backward bifurcation cases. The backward bifurcation is generated by the high sporulation of young leaves compared to mature ones. In this case, when the basic reproduction number is less than one, the disease can persist. These results provide useful insights on the disease dynamics and its control. In particular, ontogenic resistance may require higher control efforts to eradicate the disease. [ABSTRACT FROM AUTHOR]

Fick’s law and the Fokker–Planck law of diffusion are applied to manifest the cognitive dispersal of individuals in two reaction-diffusion SEIR epidemic models, where the disease transmission is illustrated by nonlocal infection mechanisms in heterogeneous environments. Building upon the well-posedness of solutions, threshold dynamics are discussed in terms of the basic reproduction numbers for the two cognitive epidemic models. The numerical investigation reveals that the Fokker–Planck law can better describe the diffusion of individuals by taking different dispersal strategies of exposed individuals in our cognitive epidemic models, and provides some insights on spatial segregation and nonpharmaceutical interventions: (i) spatial segregation occurs in the random diffusion model when the nonlocal infection radius is small, while it appears in the symmetric diffusion model when the radius is large; (ii) nonpharmaceutical interventions on restricting the dispersal of exposed and infected individuals do not contribute to reducing the infection proportion, but rather eliminate the disease in a region, which expands as the nonlocal infection radius increases. We additionally find that the final infection size in the random diffusion model is significantly smaller than that in the symmetric diffusion model and decreases as the nonlocal infection radius increases. [ABSTRACT FROM AUTHOR]

In this paper, we establish a discrete model with periodic parameters to depict the Wolbachia spread dynamics in mosquito populations in cyclic environments. This work modifies the models established in the existing literature that did not take into account the variation of parameters with environmental periodic changes due to seasonality and other factors. When the parameters in our model are constants, it has been extensively studied and widely used. We present a conjecture about the existence of at most two periodic solutions worthy of further study, and show that the conjecture is true for the special case of 2-periodic parameters. Numerical simulations are also provided to illustrate the occurrence of periodic phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive predator–prey model with two delays, i.e., a spatial memory delay and a toxic effect delay. Initially, we analyze the global existence of the solution of the system. We then analyze the equilibria’s local stability without delays and investigate the Hopf bifurcation induced by one delay. Subsequently, we establish an analytical framework for constructing the stability switching curve in the delay space. Finally, we present numerical simulations to validate the theoretical results and verify the emergence of various spatial patterns in the system. [ABSTRACT FROM AUTHOR]

We develop a multi-group and multi-patch model to study the effects of population dispersal on the spatial spread of vector-borne diseases across a heterogeneous environment. The movement of host and/or vector is described by Lagrangian approach in which the origin or identity of each individual stays unchanged regardless of movement. The basic reproduction number R 0 of the model is defined and the strong connectivity of the host-vector network is succinctly characterized by the residence times matrices of hosts and vectors. Furthermore, the definition and criterion of the strong connectivity of general infectious disease networks are given and applied to establish the global stability of the disease-free equilibrium. The global dynamics of the model system are shown to be entirely determined by its basic reproduction number. We then obtain several biologically meaningful upper and lower bounds on the basic reproduction number which are independent or dependent of the residence times matrices. In particular, the heterogeneous mixing of hosts and vectors in a homogeneous environment always increases the basic reproduction number. There is a substantial difference on the upper bound of R 0 between Lagrangian and Eulerian modeling approaches. When only host movement between two patches is concerned, the subdivision of hosts (more host groups) can lead to a larger basic reproduction number. In addition, we numerically investigate the dependence of the basic reproduction number and the total number of infected hosts on the residence times matrix of hosts, and compare the impact of different vector control strategies on disease transmission. [ABSTRACT FROM AUTHOR]

In this paper, we explore the solvability and the optimal control problem for a compartmental model based on reaction–diffusion partial differential equations describing a transmissible disease. The nonlinear model takes into account the disease spreading due to the human social diffusion, under a dynamic heterogeneity in infection risk. The analysis of the resulting system provides the existence proof for a global solution. Moreover, we discuss the conditions of optimality, which are of some help to reduce the concentration of the infected population in certain spatial areas. [ABSTRACT FROM AUTHOR]

In this paper, assuming the certain variable satisfies the Ornstein–Uhlenbeck process, we formulate a stochastic within-host dengue model with immune response to obtain further understanding of the transmission dynamics of dengue fever. Then we analyze the dynamical properties of the stochastic system in detail, including the existence and uniqueness of the global solution, the existence of a stationary distribution, and the extinction of infected monocytes and free viruses. In particular, it is worth revealing that we get the specific form of covariance matrix in its probability density around the quasi-endemic equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive susceptible-infected-susceptible epidemic model in a spatiotemporally heterogeneous environment. We consider a saturated incidence function of the form S I m + S + I , where m is a nonnegative function. When there is a positive disease-induced mortality rate everywhere, we demonstrate that the disease will always become extinct and the susceptible population will stabilize at zero or a positive constant. In the case where the disease-induced mortality rate is negligible, we examine the model in a time-periodic and spatially heterogeneous environment and establish the threshold dynamics between disease extinction and persistence in terms of the basic reproduction number. Under certain conditions, we show the global attractivity of both the disease-free equilibrium and endemic equilibrium by constructing appropriate Lyapunov functions. Moreover, we determine the spatial distribution of the disease when the diffusion rate of the susceptible or infected population (or both) is sufficiently small. Our findings suggest that the presence of a saturation effect reduces the transmission risk, allows for the total population size to have a substantial impact on disease dynamics, and may significantly alter the spatial distribution of the disease under certain circumstances. [ABSTRACT FROM AUTHOR]

Rational allocation of limited vaccine resources is one of the key issues in the prevention and control of emerging infectious diseases. An age-structured infectious disease model with limited vaccine resources is proposed to explore the optimal vaccination ages. The effective reproduction number R e of the epidemic disease is computed. It is shown that the reproduction number is the threshold value for eradicating disease in the sense that the disease-free steady state is globally stable if R e < 1 and there exists a unique endemic equilibrium if R e > 1 . The effective reproduction number is used as an objective to minimize the disease spread risk. Using the epidemic data from the early spread of Wuhan, China and demographic data of Wuhan, we figure out the strategies to distribute the vaccine to the age groups to achieve the optimal vaccination effects. These analyses are helpful to the design of vaccination schedules for emerging infectious diseases. [ABSTRACT FROM AUTHOR]

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease’s persistence or eradication depends on the basic reproduction number R 0 s . If R 0 s < 1 , the disease is eradicated with a probability of 1, while R 0 s > 1 results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if R 0 s > 1 , the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required. [ABSTRACT FROM AUTHOR]

Heterogeneity in susceptibility and infectivity is a central issue in epidemiology. Although the latter has received some attention recently, the former is often neglected in modeling of epidemic systems. Moreover, very few studies consider both of these heterogeneities. This paper is concerned with the characterization of epidemic models with differential susceptibility and differential infectivity under a general setup. Specifically, we investigate the global asymptotic behavior of equilibria of these systems when the network configuration of the Susceptible-Infectious interactions is strongly connected. These results prove two conjectures by Bonzi et al. (J Math Biol 62:39–64, 2011) and Hyman and Li (Math Biosci Eng 3:89–100, 2006). Moreover, we consider the scenario in which the strong connectivity hypothesis is dropped. In this case, the model exhibits a wider range of dynamical behavior, including the rise of boundary and interior equilibria, all based on the topology of network connectivity. Finally, a model with multidirectional transitions between infectious classes is presented and completely analyzed. [ABSTRACT FROM AUTHOR]

A characteristic of malaria in all its forms is the potential for superinfection (that is, multiple concurrent blood-stage infections). An additional characteristic of Plasmodium vivax malaria is a reservoir of latent parasites (hypnozoites) within the host liver, which activate to cause (blood-stage) relapses. Here, we present a model of hypnozoite accrual and superinfection for P. vivax. To couple host and vector dynamics for a homogeneously-mixing population, we construct a density-dependent Markov population process with countably many types, for which disease extinction is shown to occur almost surely. We also establish a functional law of large numbers, taking the form of an infinite-dimensional system of ordinary differential equations that can also be recovered by coupling expected host and vector dynamics (i.e. a hybrid approximation) or through a standard compartment modelling approach. Recognising that the subset of these equations that model the infection status of the human hosts has precisely the same form as the Kolmogorov forward equations for a Markovian network of infinite server queues with an inhomogeneous batch arrival process, we use physical insight into the evolution of the latter process to write down a time-dependent multivariate generating function for the solution. We use this characterisation to collapse the infinite-compartment model into a single integrodifferential equation (IDE) governing the intensity of mosquito-to-human transmission. Through a steady state analysis, we recover a threshold phenomenon for this IDE in terms of a parameter R 0 expressible in terms of the primitives of the model, with the disease-free equilibrium shown to be uniformly asymptotically stable if R 0 < 1 and an endemic equilibrium solution emerging if R 0 > 1 . Our work provides a theoretical basis to explore the epidemiology of P. vivax, and introduces a strategy for constructing tractable population-level models of malarial superinfection that can be generalised to allow for greater biological realism in a number of directions. [ABSTRACT FROM AUTHOR]

This paper studies a two-age structured SIRS epidemic model with logistic growth of susceptible population and two-time delays. We simultaneously introduce two-time delays, i.e., the immunity and incubation periods, into this dynamic system and investigate their impact on different dynamic behaviors for the model. By means of the C 0 -semigroup theory, the model is transformed into a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the endemic equilibrium is obtained. Following the spectral analysis, the characteristic equation technique, and the Hopf bifurcation theorem, we show that different combinations of the two delays perform a vital role in the instability/stability as well as the Hopf bifurcation results of equilibrium solutions. We numerically provide some graphical representations to check the main theoretical results and show the rich dynamics by varying the two delay parameters. [ABSTRACT FROM AUTHOR]

The region of St. Louis, Missouri, has displayed a high level of heterogeneity in COVID-19 cases, hospitalization, and vaccination coverage. We investigate how human mobility, vaccination, and time-varying transmission rates influenced SARS-CoV-2 transmission in five counties in the St. Louis area. A COVID-19 model with a system of ordinary differential equations was developed to illustrate the dynamics with a fully vaccinated class. Using the weekly number of vaccinations, cases, and hospitalization data from five counties in the greater St. Louis area in 2021, parameter estimation for the model was completed. The transmission coefficients for each county changed four times in that year to fit the model and the changing behaviour. We predicted the changes in disease spread under scenarios with increased vaccination coverage. SafeGraph local movement data were used to connect the forces of infection across various counties. [ABSTRACT FROM AUTHOR]

Ehrlichia chaffeensis is a tick-borne disease transmitted by ticks to dogs. Few studies have mathematical modelled such tick-borne disease in dogs, and none have developed models that incorporate different ticks' developmental stages (discrete variable) as well as the duration of infection (continuous variable). In this study, we develop and analyze a model that considers these two structural variables using integrated semigroups theory. We address the well-posedness of the model and investigate the existence of steady states. The model exhibits a disease-free equilibrium and an endemic equilibrium. We calculate the reproduction number ( $ \mathcal {T}_0 $ T 0 ). We establish a necessary and sufficient condition for the bifurcation of an endemic equilibrium. Specifically, we demonstrate that a bifurcation, either backward or forward, can occur at $ \mathcal {T}_0=1 $ T 0 = 1 , leading to the existence, or not, of an endemic equilibrium even when $ \mathcal {T}_0 \lt 1 $ T 0 < 1. Finally, numerical simulations are employed to illustrate these theoretical findings. [ABSTRACT FROM AUTHOR]

We consider a Darwinian (evolutionary game theoretic) version of a standard susceptible-infectious SI model in which the resistance of the disease causing pathogen to a treatment that prevents death to infected individuals is subject to evolutionary adaptation. We determine the existence and stability of all equilibria, both disease-free and endemic, and use the results to determine conditions under which the treatment will succeed or fail. Of particular interest are conditions under which a successful treatment in the absence of resistance adaptation (i.e. one that leads to a stable disease-free equilibrium) will succeed or fail when pathogen resistance is adaptive. These conditions are determined by the relative breadths of treatment effectiveness and infection transmission rate distributions as functions of pathogen resistance. [ABSTRACT FROM AUTHOR]

Dengue fever creates more than 390 million cases worldwide yearly. The most effective way to deal with this mosquito-borne disease is to control the vectors. In this work we consider two weapons, the endosymbiotic bacteria Wolbachia and predators of mosquito larvae, for combating the disease. As Wolbachia-infected mosquitoes are less able to transmit dengue virus, releasing infected mosquitoes to invade wild mosquito populations helps to reduce dengue transmission. Besides this measure, the introduction of predators of mosquito larvae can control mosquito population. To evaluate the impact of the predators on Wolbachia spreading dynamics, we develop a stage-structured five-dimensional model, which links the predator-prey dynamics with the Wolbachia spreading. By comparatively analysing the dynamics of the models without and with predators, we observe that the introduction of the predators augments the number of coexistence equilibria and impedes Wolbachia spreading. Some numerical simulations are presented to support and expand our theoretical results. [ABSTRACT FROM AUTHOR]

We propose a hand, foot and mouth disease (HFMD) transmission model for children with behaviour change and imperfect quarantine. The symptomatic and quarantined states obey constant behaviour change while others follow variable behaviour change depending on the numbers of new and recent infections. The basic reproduction number $ \mathcal {R}_0 $ R 0 of the model is defined and shown to be a threshold for disease persistence and eradication. Namely, the disease-free equilibrium is globally asymptotically stable if $ \mathcal {R}_0\le 1 $ R 0 ≤ 1 whereas the disease persists and there is a unique endemic equilibrium otherwise. By fitting the model to weekly HFMD data of Shanghai in 2019, the reproduction number is estimated at 2.41. Sensitivity analysis for $ \mathcal {R}_0 $ R 0 shows that avoiding contagious contacts and implementing strict quarantine are essential to lower HFMD persistence. Numerical simulations suggest that strong behaviour change not only reduces the peak size and endemic level dramatically but also impairs the role of asymptomatic transmission. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

We develop, in this paper, a reaction–diffusion epidemic model with the general incidence rate and personal protection awareness, where, the direct transmission and indirect transmission are also introduced to describe the complexity of the spread of infectious diseases. The existence, uniqueness and boundedness of global solution are obtained first. And then, the basic reproduction number R 0 is derived to go as the threshold value for predicting the persistence and extinction of disease. To be more specific, model admits a globally asymptotically stable disease-free steady state if R 0 < 1 , while the disease is persistent for R 0 > 1 . Further, for R 0 = 1 , the global asymptotic stability of the disease-free steady state is demonstrated when all diffusion coefficients are constants. In addition, the global asymptotical stability of the endemic equilibrium of this model with spatially homogeneous is derived if the corresponding basic reproduction is great than 1. Particularly, we also discuss the existence of traveling wave solution and the minimum wave speed c ∗ of the spatial homology model. Numerical simulations are presented to validate the main results and provide some recommendations for the control of multi-transmission diseases. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the existence and nonexistence of traveling wave solution for a diffusive vector-borne disease model with nonlocal reaction and distributed delays. We demonstrate that the basic reproduction number R 0 of the corresponding ordinary differential equation system as a threshold determines whether the model admits traveling waves or not and there exists a critical wave speed c m ∗ > 0 when R 0 > 1 . Specifically, (i) As R 0 > 1 and the wave speed c > c m ∗ , the existence of traveling waves for the system is established with the aid of a perturbed system; (ii) As R 0 > 1 and 0 < c < c m ∗ , the nonexistence of traveling waves is proved via the two-sided Laplace transform; (iii) As R 0 ≤ 1 and c > 0 , the nonexistence is obtained by utilizing the comparison principle. The theoretical results are applied to dengue fever epidemics. We study the effects of geographical movement, nonlocal interaction, incubation period and R 0 on the threshold speed c m ∗ for dengue fever. [ABSTRACT FROM AUTHOR]

In this paper, we will revisit the model studied in Lou and Zhao (J Math Biol 62:543–568, 2011), where the model takes the form of a nonlocal and time-delayed reaction–diffusion model arising from the fixed incubation period. We consider the infection age to be a continuous variable but without the limitation of the fixed incubation period, leading to an age-space structured malaria model in a bounded domain. By performing the elementary analysis, we investigate the well-posedness of the model by proving the global existence of the solution, define the explicit formula of basic reproduction number when all parameters remain constant. By analyzing the characteristic equations and designing suitable Lyapunov functions, we also establish the threshold dynamics of the constant disease-free and positive equilibria. Our theoretical results are also validated by numerical simulations for 1-dimensional and 2-dimensional domains. [ABSTRACT FROM AUTHOR]

The disease-induced herd immunity level h D is the fraction of the population that must be infected by an epidemic to ensure that a new epidemic among the remaining susceptible population is not supercritical. For a homogeneously mixing population h D equals the classical herd immunity level h C , which is the fraction of the population that must be vaccinated in advance of an epidemic so that the epidemic is not supercritical. For most forms of heterogeneous mixing h D < h C , sometimes dramatically so. For an SEIR (susceptible → exposed → infective → recovered) model of an epidemic among a population that is partitioned into households, in which individuals mix uniformly within households and, in addition, uniformly at a much lower rate in the population at large, we show that h D > h C unless variability in the household size distribution is sufficiently large. Thus, introducing household structure into a model typically has the opposite effect on disease-induced herd immunity than most other forms of population heterogeneity. We reach this conclusion by considering an approximation h ~ D of h D , supported by numerical studies using real-world household size distributions. For n = 2 , 3 , we prove that h ~ D > h C when all households have size n, and conjecture that this inequality holds for any common household size n. We prove results comparing h ~ D and h C for epidemics which are highly infectious within households, and also for epidemics which are weakly infectious within households. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with two SIS epidemic reaction–diffusion models with mass action infection mechanism of the form SI, and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small. For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval. In sharp contrast, for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals, our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant, and the infected population may concentrate only at some isolated highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat, depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations. Numerical simulations are performed to support and complement our theoretical findings. [ABSTRACT FROM AUTHOR]

Almost all models used in analysis of infectious disease outbreaks contain some notion of population size, usually taken as the census population size of the community in question. In many settings, however, the census population is not equivalent to the population likely to be exposed, for example if there are population structures, outbreak controls or other heterogeneities. Although these factors may be taken into account in the model: adding compartments to a compartmental model, variable mixing rates and so on, this makes fitting more challenging, especially if the population complexities are not fully known. In this work we consider the concept of effective population size in outbreak modelling, which we define as the size of the population involved in an outbreak, as an alternative to use of more complex models. Effective population size is an important quantity in genetics for estimation of genetic diversity loss in populations, but it has not been widely applied in epidemiology. Through simulation studies and application to data from outbreaks of COVID-19 in China, we find that simple SIR models with effective population size can provide a good fit to data which are not themselves simple or SIR. [ABSTRACT FROM AUTHOR]

Understanding both the epidemiological and evolutionary dynamics of antimicrobial resistance is a major public health concern. In this paper, we propose a nested model, explicitly linking the within- and between-host scales, in which the level of resistance of the bacterial population is viewed as a continuous quantitative trait. The within-host dynamics is based on integro-differential equations structured by the resistance level, while the between-host scale is additionally structured by the time since infection. This model simultaneously captures the dynamics of the bacteria population, the evolutionary transient dynamics which lead to the emergence of resistance, and the epidemic dynamics of the host population. Moreover, we precisely analyze the model proposed by particularly performing the uniform persistence and global asymptotic results. Finally, we discuss the impact of the treatment rate of the host population in controlling both the epidemic outbreak and the average level of resistance, either if the within-host scale therapy is a success or failure. We also explore how transitions between infected populations (treated and untreated) can impact the average level of resistance, particularly in a scenario where the treatment is successful at the within-host scale. [ABSTRACT FROM AUTHOR]

This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem is formulated in terms of an invariant which is similar to the basic reproduction number of a related deterministic model. Compared to the deterministic model, stability of the disease-free equilibrium holds more generally for the stochastic model. The optimal control theory is applied to the stochastic model, revealing some important new insights. Theoretical results are illustrated by way of simulations. [ABSTRACT FROM AUTHOR]

In this paper, we establish a discrete model with periodic parameters to depict the Wolbachia spread dynamics in mosquito populations in cyclic environments. This work modifies the models established in the existing literature that did not take into account the variation of parameters with environmental periodic changes due to seasonality and other factors. When the parameters in our model are constants, it has been extensively studied and widely used. We present a conjecture about the existence of at most two periodic solutions worthy of further study, and show that the conjecture is true for the special case of 2-periodic parameters. Numerical simulations are also provided to illustrate the occurrence of periodic phenomena. [ABSTRACT FROM AUTHOR]